Can you transform the square shown below into pieces that when put together form five small equal squares whose total area shall be equal to the square down below with exactly four cuts?

How about this (see attached image):

Use the four cuts as shown in the image (red lines).

Then assemble 5 equal squares by the numbers: 1 center square and the rest are pieced together using two pieces as shown. All five together add up to the same area as original square because we use all pieces.

The way one gets a hint toward a solution is to see how an area of a square of length 1 can be split into 5 equal square areas:

which indicates we need to find a a triangle with sides 2 and 1 to get the hypotenuse of the right length. That gave rise to the cut pattern (if you look carefully, there are triangles with those side lengths).